I have a 1 gallon aquarium, with a filter and a heater. I do not have a water tester, but the water is really clear and my betta (Mr. Wubbers) is very active. My question is do I need to change the water more than once a week? At first the water was cloudy but like I said it is really clear.

Clear water does not equal good water, you can't see ammonia, nitrites or nitrates. On a tank that size I wouldn't bother with a filter & do 2-3 water changes per week, one of which is 100%. I do 2 50%+ water changes on my 2.5g tanks with a 100% every 2-3 wks. I would get a liquid test kit & see how fast the ammonia, nitrites & nitrates build up. If your tank is new its not cycled & you need to do very frequent water changes until it is cycled if it cycles.